Higher-Order Finite Element Solutions of Option Prices

Kinks and jumps in the payoff function of option contracts prevent an effective
implementation of higher-order numerical approximation methods. Moreover, the
derivatives (the greeks) are not easily determined around such singularities, even with
standard lower-order methods. This paper suggests a transformation to turn the original
ill-conditioned pricing problem into a well-behaved numerical problem. For a
standard test case, both vanilla- and binary call price functions are approximated with
(tensor) B-splines of up to 10’th order. Polynomial convergence rates of orders up to
approximately 10 are obtained for prices as well as for first and second order derivatives
(delta and gamma). Unlike similar studies, numerical approximation errors are
measured both as weighted averages and in the supnorm over a state space including
time-to-maturities down to a split second.
KEYWORDS: Numerical option pricing, Transformed state spaces, Higher-order
B-splines.